Question

the limitations of logistic function for the spread of information

Answer 1

Answer:

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Step-by-step explanation:

One clever example of logistic growth is the spreading of a rumor in a population. Suppose that one person knows a secret, and once a day, anyone who knows the secret can share it with one other person, but without knowing whether that person already knows it. Well, early on, it's unlikely that a teller will run across someone who already knows the secret, but later, when more people know, it's less likely to find a person who doesn't know.

In this example, I assumed we have a group of 20 people, and that person #1 knows the secret to begin with. Then, on each "round," I generated a random number (using a spreadsheet) between 1 and 20, to choose whom to tell the secret next. the results are in the table below.

 

In round one, 1 told 4. In round two, 1 and 4 told 20 and 6, for a total of four secret-knowers. In round three, those four told four new people to increase the total to 8.

But notice that in round four, we begin to tell people who already know the secret, so the accumulation of secret knowers begins to slow down.

That continues until poor #14 finally learns the secret after eight rounds. The results are plotted here and you can see that it's just like our logistic growth curves.

Limits to growth

One important feature of the logistic function is it's behavior at large values of the independent variable. Here we'll define a population function n(t) as a logistic function.

n(t)=L1+e−k(t−to)

There are two adjustable parameters in this function, L and k. These are a vertical scaling parameter ( L ) and a horizontal scaling parameter ( k ) that allow us to stretch or compress such a function to fit our data.

We're interested in the limiting behavior as our variable t (for time) increases to infinity:

 

limt→∞n(t)=L1+e−1k(t−to)

Now if we look at the part of the function that contains t, and replace the negative exponential with a fraction, we see that the fraction tends toward zero as t gets large because 1 over a very large number tends toward zero as the denominator grows:

limt→∞1ek(t−to)=0

 

So the overall limit of the function as t gets very large (after a sufficient time has passed) is

limt→∞L1+e−1k(t−to)

So the limit of the function is the numerator, L. This makes the limiting value of a logistic function easy to find, and it makes a logistic function relatively easy to write given enough data.

Answer 2

Step-by-step explanation:

The logistic function models the exponential growth of a population, but also considers factors like the carrying capacity of land.

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